Geometric vector bundles/Locally free sheaves/Correspondence/Section

Definition

Let ${}X$ denote a scheme. A scheme ${}V$ equipped with a morphism

$p\colon V\longrightarrow X$

is called a geometric vector bundle of rank ${}r$ over ${}X$ if there exists an open covering ${}X=\bigcup _{i\in I}U_{i}$ and ${}U_{i}$-isomorphisms

$\psi _{i}\colon U_{i}\times {{\mathbb {A} }_{}^{r}}={{\mathbb {A} }_{U_{i}}^{r}}\longrightarrow V{|}_{U_{i}}=p^{-1}(U_{i})$

such that for every open affine subset ${}U\subseteq U_{i}\cap U_{j}$, the transition mappings

$\psi _{j}^{-1}\circ \psi _{i}\colon {{\mathbb {A} }_{U_{i}}^{r}}{|}_{U}\longrightarrow {{\mathbb {A} }_{U_{j}}^{r}}{|}_{U}$

are linear automorphisms, i.e. they are induced by an automorphism of the polynomial ring ${}\Gamma (U,{\mathcal {O}}_{X})[T_{1},\ldots ,T_{r}]$ given by ${}T_{i}\mapsto \sum _{j=1}^{r}a_{ij}T_{j}$.

Here we can restrict always to affine open coverings. If ${}X$ is separated then the intersection of two affine open subschemes is again affine and then it is enough to check the condition on the intersections. The trivial bundle of rank ${}r$ is the ${}r$-dimensional affine space ${}{{\mathbb {A} }_{X}^{r}}$ over ${}X$, and locally every vector bundle looks like this. Many properties of an affine space are enjoyed by general vector bundles. For example, in the affine space we have the natural addition

$+\colon {{\mathbb {A} }_{U}^{r}}\times _{U}{{\mathbb {A} }_{U}^{r}}\longrightarrow {{\mathbb {A} }_{U}^{r}},(v_{1},\ldots ,v_{r},w_{1},\ldots ,w_{r})\longmapsto (v_{1}+w_{1},\ldots ,v_{r}+w_{r}),$

and this carries over to a vector bundle, that is, we have an addition

$\alpha \colon V\times _{X}V\longrightarrow V.$

The reason for this is that the isomorphisms occurring in the definition of a geometric vector bundle are linear, hence the addition on ${}V{|}_{U}$ coming from an isomorphism with some affine space over ${}U$ is independent of the choosen isomorphism. For the same reason there is a unique closed subscheme of ${}V$ called the zero-section which is locally defined to be ${}0\times U\subseteq {{\mathbb {A} }_{U}^{r}}$. Also, multiplication by a scalar, i.e. the mapping

$\cdot \colon {\mathbb {A} }_{U}^{1}\times _{U}{{\mathbb {A} }_{U}^{r}}\longrightarrow {{\mathbb {A} }_{U}^{r}},(s,v_{1},\ldots ,v_{r})\longmapsto (sv_{1},\ldots ,sv_{r}),$

carries over to a scalar multiplication

$\cdot \colon {\mathbb {A} }_{X}\times _{X}V\longrightarrow V.$

In particular, for every point ${}P\in X$ the fiber ${}V_{P}=V\times _{X}P$ is an affine space over ${}\kappa (P)$.

For a geometric vector bundle ${}p\colon V\rightarrow X$ and an open subset ${}U\subseteq X$ one sets

${}\Gamma (U,V)={\left\{s:U\rightarrow V{|}_{U}\mid p\circ s=\operatorname {Id} _{U}\right\}}\,,$

so this is the set of sections in ${}V$ over ${}U$. This gives in fact for every scheme over ${}X$ a set-valued sheaf. Because of the observations just mentioned, these sections can also be added and multiplied by elements in the structure sheaf, and so we get for every vector bundle a locally free sheaf, which is free on the open subsets where the vector bundle is trivial.

Definition

A coherent ${}{\mathcal {O}}_{X}$-module ${}{\mathcal {F}}$ on a scheme ${}X$ is called locally free of rank ${}r$, if there exists an open covering ${}X=\bigcup _{i\in I}U_{i}$ and ${}{\mathcal {O}}_{U_{i}}$-module-isomorphisms ${}{\mathcal {F}}{|}_{U_{i}}\cong {\left({\mathcal {O}}_{U_{i}}\right)}^{r}$ for every

${}i\in I$.

Vector bundles and locally free sheaves are essentially the same objects.

Theorem

Let ${}X$ denote a scheme. Then the category of locally free sheaves on ${}X$ and the category of geometric vector bundles on ${}X$ are equivalent. A geometric vector bundle ${}V\rightarrow X$ corresponds to the sheaf of its sections, and a locally free sheaf ${}{\mathcal {F}}$ corresponds to the (relative)

spectrum of the symmetric algebra of the dual module ${}{\mathcal {F}}^{*}$.

The free sheaf of rank ${}r$ corresponds to the affine space ${}{{\mathbb {A} }_{X}^{r}}$ over ${}X$.